Taylor polynomials: Sine function

As in the case of the Exponential function, the parabolas give usable approximations to the original curve for a greater and greater interval. The parabolas make the effort to share more and more oscillations with the sine curve. (Felix Klein)

Taylor's series of the sine function at x=0 is:

We say that the sine function is an odd function because in this power series (and in the Taylor's polynomials centered at the origin) the only terms are those with odd exponent.

We can change the center and see that the behavior of the approximation is equally good.

The behavior of these power series is very good. We say then that the infinite series for the sine function converge for all values of x.

It can't be the case for all functions. For example, what happens if some values aren't in the Domain of a function?. We can study one simple example: Taylor polynomials (3): Square root.

REFERENCES

Felix Klein - Elementary Mathematics from an Advanced Standpoint. Arithmetic, Algebra, Analysis (pags. 223-228) - Dover Publications

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The function is not defined for values less than -1. Taylor polynomials about the origin approximates the function between -1 and 1.

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By increasing the degree, Taylor polynomial approximates the exponential function more and more.